A Decomposed Genetic Algorithm for Solving the Joint Product Family Optimization Problem

7/29/2019 A Decomposed Genetic Algorithm for Solving the Joint Product Family Optimization Problem

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American Institute of Aeronautics and Astronautics

1

A Decomposed Genetic Algorithm for Solving the Joint

Product Family Optimization Problem

Aida Khajavirad* and Jeremy J. Michalek

Carnegie Mellon University, Pittsburgh, PA 15213 USA

Timothy W. SimpsonThe Pennsylvania State University, University Park, PA, 16802 USA

A critical step when designing a successful product family is to determine a cost-saving

platform configuration along with an optimally distinct set of product variants that target

different market segments. Numerous optimization-based approaches have been proposed to

help resolve the tradeoff between platform commonality and the ability to achieve distinct

performance targets for each variant. However, the high dimensionality of an all-in-one

algorithm for optimizing the joint problem of 1) platform variable selection, 2) platform

design and 3) variant design makes most of these approaches impractical when a large

number of products is considered. Many existing approaches have restricted the scope of theproblem by fixing platform configuration a priori, limiting platform configuration to an all-

or-none component sharing strategy, or by solving subsets of the joint problem in stages,

sacrificing optimality. In this study, we propose a single-stage optimization approach for

solving the joint product family problem with generalized commonality using an efficient

decomposition solution strategy involving multi-objective genetic algorithms (MOGAs). The

proposed approach overcomes prior limitations by introducing a generalized two-

dimensional commonality chromosome and decomposing the joint formulation into a two-

level GA, where the upper-level determines the optimal platform configuration while each

lower-level designs one of the individual variants in the family. Moreover, all sub-problems

run in parallel, and the upper-level GA coordinates consistency among the lower-levels using

the MPI (Message Passing Interface) library. The proposed approach is demonstrated by

optimizing a family of three general aviation aircraft, and results outperform those from a

non-decomposed GA. Results also show that the commonality-performance Pareto frontcontains solutions with generalized commonality, suggesting the need to avoid all-or-none

component sharing restrictions in order to avoid sub-optimality. Future work in scaling the

decomposed GA to larger product families is also discussed.

I. IntroductionARKETPLACE globalization, the proliferation of niche markets driven by heterogeneity of preferences,

increased competitive pressures, and demand for differentiated and customized products have rendered the

practice of isolated design and production of individual products nearly obsolete. Across many industries, the

prevailing practice is to design lines or families of product variants that exploit commonality to take advantage of

economies of scale and scope while targeting a variety of market segments and achieving strategic market coverage

to deter competitors. Planning families of products requires particular care and attention, since each productcompetes for market share not only with competitor products, but also with other products in the family.

Generally speaking, a product family is a group of related products (i.e., variants) that are derived from a

common set of components, modules, and/or subsystems to satisfy a variety of market niches, and the key to a

successful product family is theproduct platform around which the product family is derived1. Designing a product

* Graduate Research Assistant, Mechanical Engineering, Student Member AIAA, Email: [email protected] Assistant Professor, Mechanical Engineering, Member AIAA, Email: [email protected]. Professor of Mechanical Engineering and Industrial Engineering, Senior Member AIAA, Email: [email protected].

M


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platform and corresponding family of products is a difficult task that embodies all of the challenges of product

design while adding the complexity of coordinating the design of multiple products in an effort to increase

commonality across the set of products without compromising their individual performance. This challenge

manifests early in the design process wherein designers must not only specify the platform configuration (i.e.,

selecting which design variables are shared across the products in the family also referred to as platform variable

selection orplatform selection2), but also optimize the design of the platform and the individual variants by choosing

design variable values while maintaining commonality defined in the platform configuration. Resolving the inherent

tradeoff between platform commonality and product distinctiveness is paramount: Increasing the degree of

commonality among variants in a product family generally reduces total cost, but it can also compromise the ability

of each variant to fully achieve the desired characteristics that make it distinct and attractive to different market

segments.

In the next section, we review related work in design optimization that has been developed to map out this

tradeoff between commonality and distinctiveness. In Section III we propose an all-in-one MOGA for solving the

joint product family problem with a generalized commonality chromosome that allows commonality among subsets

of the variants. In Section IV we decompose the GA to dramatically improve search efficiency and scalability by

reducing the search space of each sub-GA and enabling use of parallel processing. In Section V, an example

involving the design of a family of general aviation aircraft is presented and optimized using both sequential and

parallel algorithms for comparison. Closing remarks and future work are given in the final section.

II. Review of Related LiteratureNumerous optimization approaches have been developed within the engineering design community during the

past decade to help solve the product family design problem. Simpson3 reviews and classifies 40 approaches from

the literature. In many of these approaches, the design variables that define the product platform within the family

are known or specified a priori, i.e., before performing the optimization (Allada and Jiang4, Blackenfelt5, Chang and

Ward6, Dsouza and Simpson7, Dai and Scott8, Farrell and Simpson9, Fellini et al.10, Fujita et al.11, Gonzales-Zugasti

et al.12,13, Hernandez et al.14, Kokkolaras et al.15, Kumar et al.16, Li and Azarm17, Messac et al.18, Nelson et al.19,

Ortega et al.20, Seepersad et al.21,22, Simpson et al.23,24), whereas in other instances, the platform variable selection is

determined during optimization, i.e., the platform is specified a posteriori (Akundi et al.25, Cetin and Saitou26, Dai

and Scott8

, de Wecket al.27

, Fellini et al.28,29

, Fujita and Yoshida30

, Fujita et al.31

, Gonzales-Zugasti and Otto32

,Hernandez et al.33,34, Messac et al.35, Nayaket al.36, Rai and Allada37, Hassan et al.38, Simpson and Dsouza7, Khire

et al.2). In a similar manner, Fujita39 has divided product variety optimization problems into three classes: In Class-I

problems, product attributes are optimized under a fixed platform configuration (i.e., the platform is known a

priori); Class-II problems deal with finding the optimal module selection using predefined modules (i.e., the design

of each module is known a priori); and finally, in Class-III problems, the product attributes and platform

configuration are optimized simultaneously. We refer to this Class III, a posteriori problem as the joint product

family optimization problem because it involves determining the optimal combination of 1) platform variable

selection, 2) platform design and 3) variant design. Since each of these decisions is generally dependent on the

others, approaches with restricted scope cannot guarantee optimality, except in special cases, and there is a clear

need for an algorithm capable of solving the joint problem for practical product family applications.

Simpson3 classifies optimization-based approaches for solving the product family design problem where

platform variable selection is specified a posteriori based on the number of stages involved: single-stage approachesoptimize the product platform and resulting family of products simultaneously (Akundi et al.25, Cetin and Saitou26,

Fujita et al.31, Fujita and Yoshida30, Gonzales-Zugasti and Otto32, Hassan et al.38, Simpson Dsouza7, Khire et al.2)

while two-stage approaches optimize the platform variables first and then instantiate the individual variants based

on this platform during the second stage of the optimization (Dai and Scott8, De Wecket al.27, Hernandez et al.33,34,

Messac et al.35, Nayaket al.36, Rai and Allada37). A special case is Fellini et al.28,29, who solve the joint problem by

specifying platform variable selection in the first stage and optimizing the platform and variant variables in the

second stage. While both single- and two-stage approaches can be effective at determining design variable settings


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for the product family, two-stage approaches may lead to sub-optimal solutions in general, therefore single-stage

approaches are preferred on the criterion of optimality. A major disadvantage of single-stage approaches, however,

is that the dimensionality of the optimization problem is considerably higher compared to two-stage approaches,

which makes these approaches impractical when designing a family with a large number of products. In summary,

an efficient approach is needed for solving the joint product family optimization problem in a single-stage while

effectively handling problems involving large numbers of products.

Among prior product family optimization approaches, many studies have found that Genetic Algorithms (GAs)

are well-suited for optimizing a product platform and its corresponding family of products. Li and Azarm17 were

among the first in the engineering design community to use GAs to solve the product family design problem, but

they required specification of the platform a priori. DSouza and Simpson40 developed a similar approach using the

Non-Dominated Sorting GA (NSGA-II)41, which was then extended using an augmented chromosome string to

examine varying levels of platform commonality to simultaneous determine platform variable selection and design

of the platform and variants.7 In particular, each chromosome string was augmented by n additional commonality

controlling genes, where n is the number of candidate platform variables. Each of these commonality variables can

take the value of either 0 or 1, where a value of 1 means that the corresponding design variable is made common

among all of the products in the family, while a value of 0 allows design variables to be unique among products. A

commonality metric was defined by summation of design variables variations within the product family. According

to this definition, the commonality in the resulting product family is based not only on how many variables arecommon but also on how similar the values of unique variables are to one another. Total deviation from each

product performance target and the commonality metric were treated as separate objective functions. This approach

was then applied to the design of a family of three general aviation aircraft, as defined by their seating capacity.

Results successfully showed the tradeoff between commonality and performance in the product family. Finally,

Hassan et al.38 used theN-Branch Tournament Selection GA for optimizing a family of three commercial satellites.

They imposed no explicit representation for controlling commonality within the family, and the chromosome

included only three sub-strings, each one representing the design variables for a specific satellite in the product line.

The commonality metric was evaluated based on the number of common design variables shared by all three sub-

strings. The problem had four objectives: the launch mass of each satellite in the product line and the commonality

metric. However, all these prior approaches have two primary limitations that restrict their scope of applicability: 1)

They restrict commonality decisions to a single platform, and 2) they have limited scalability for solving large

problems.The first restriction, which limits commonality decisions for each variable to be either a full platform variable

shared by all variants or a non-shared variable that is different among all variants, is useful for simplicity but causes

unnecessary restriction in practice: There are many cases where allowing commonality among a subset of variants

may lead to more efficient design opportunities. To overcome this limitation, we generalize Simpson and

DSouzas7 commonality controlling gene by introducing a two-dimensional chromosome that accounts for

commonality of each module among any subset of the variants.

The second limitation of the aforementioned approaches is limited scalability to large problems with many

product variants. As the number of variants increases, the chromosome string becomes long, and solving a high-

dimensional space with a single GA can be difficult, if not impossible. For instance, Akundi and Simpson 25 found

that as many as 25,000 generations with population sizes of 1,500 were needed to obtain a good spread of

solutions for a family of 10 universal motors that had three conflicting objectives, where each motor was defined by

8 design variables. The addition of generalized commonality in our formulation also adds significant complexity,making an all-in-one approach intractable in many cases. Therefore, in order to create a general and scalable

algorithm that can be applied to a large number of products without losing performance, we propose a method for

decomposing the initial all-in-one GA into an upper-level GA that controls commonality decisions and a set of

lower-level GAs: one for the design variables of each product. Each lower-level GA deals only with an individual

product and its chromosomes consist only of the design variables for that product. Commonality constraints are

imposed from the upper-level GA to the lower-level ones to enforce commonality decisions on the values of design

variables across products. Moreover, the decomposed model can be parallelized by executing each GA on a separate


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processor and using the MPI (Message Passing Interface) library for exchanging data among sub-problems. Hence,

in addition to improved performance due to using decomposition, it is possible to achieve a dramatic reduction in

computational time using parallel processing. In the next section we describe the all-in-one MOGA with the

generalized commonality chromosome, and we decompose the GA in Section IV.

III. Proposed ApproachIn this paper, we introduce a MOGA formulation for determining the Pareto front representing the tradeoff

between commonality and individual variant performance in the family. The underlying algorithm for our MOGA

code is the elitist non-dominated sorting GA (NSGA-II) introduced by Deb41, which has been shown to be capable

of finding a well-converged and well-distributed set of Pareto optimal solutions in a reasonable computational time

for many problems. In this algorithm, the offspring population Q ofN individuals is first created from the parent

population P using roulette-wheel selection, simulated binary cross-over and polynomial mutation operators. Then P

and Q are combined together to form a population R composed of 2N individuals (R=PQ). A non-dominated

sorting procedure is used to classify and sort the individuals in R into successive non-dominated fronts. The first

front is the set of individuals for which no other individual in the population has a higher fitness value for all

objectives; the second front is the non-dominated set of the remaining designs that are not in the first front; and so

on. Finally, individuals are copied into a new population Sof size N starting with the first non-dominated front,continuing with the second non-dominated front, and so on. Because R is twice as big as S, Swill not include all

design points. When the last allowed front is considered, there may exist more solutions in the last front than

remaining slots in the new population. Instead of arbitrarily discarding some members from the last front, the points

that reside in the least-crowded region in that front (having a larger crowding distance) are chosen to encourage

diversity. Using the concepts of non-dominated sorting and crowding distance in both selection and replacement

schemes makes this algorithm efficient in finding a well-distributed Pareto optimal front. However, in order to apply

the original NSGA-II code to the product family problem we have modified the chromosome representation,

crossover, and mutation operators as described in the sections that follow.

A. Chromosome RepresentationAs mentioned in Section II, we generalize the commonality controlling gene approach of Simpson and DSouza7

to relax the all-or-none component sharing restriction so that platform variables can be shared among any subset ofproduct variants. This generalization was achieved by introducing two parallel chromosomes for each individual in

the MOGA population (see ). In this representation, the first chromosome, which we call the commonality

chromosome, is a two-dimensional chromosome that defines the platform configuration. The second chromosome

contains design variables of all variants in the family and defines the optimal values of each individual product

design variables. The algorithm ensures that the two chromosomes remain consistent during evolution using

constraints imposed by the commonality chromosome on the design variable chromosome. Hence, in a product

family with p products, each defined by n design variables, the commonality chromosome is a two dimensional

matrix with p rows and n columns while the design variable chromosome contains np genes. The commonality

chromosome is generated so that genes can take any integer value between 1 and p, where any two equal integer

values for the same variable indicate that the corresponding design variables are common. An example of this

representation for a product family with three products and six design variables is shown in .


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x1 x2 x3 x4 x5 x6

Product 1 1 2 3 1 2 1

Product 2 2 2 3 2 3 3

Product 3 3 1 3 1 1 3

x1 is distinct in each product.x2 is shared between 1

st and 2nd products.x3 is shared among all products.x4 is shared between 1

st and 3rd products.x

5is distinct in each product.

x6 is shared between 2nd and 3rd products.

(a) Commonality chromosome

x11 c2 c3 c4 x51 x61 x12 c2 c3 x42 x52 c6 x13 x23 c3 c4 x53 c6

Design Variable Values Design Variable Values Design Variable Values

for Product 1 for Product 2 for Product 3

(b) Design variable chromosome

Figure 1. Two Parallel Chromosomes for Each Product Family in the GA Population

B. Crossover OperatorsDue to the 2-D configuration of the commonality chromosome, a two-dimensional binary crossover operator

was developed, which is a direct extension of the one-point crossover operator to two dimensions. In this operator,

two random integer numbers are generated in the range of (1, p) and (1, n) to select crossover sites along p and n,

wherep and n again represent the number of products and number of design variables in each product, respectively.

These two random numbers are used to divide the commonality chromosome into four quadrants. Then a third

random integer number, in the range of [1, 4], is generated to decide which quadrant is to be interchanged (see ).

Figure 2. Two-Dimensional Binary Crossover Operator

The crossover type applied to the design variable chromosome is the default used in the original NSGA-II code,

which is simulated binary crossover. After applying crossover to both chromosomes, the algorithm modifies the

design variable chromosome based on the constraints that the commonality chromosome imposes on it. In our

implementation, we implement this consistency imposition by copying the corresponding value of one of the design


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variables onto the others; however, it is possible to use alternative approaches, such as averaging the values across

variables and assigning the average value to all common variables.

C. Mutation OperatorsThe mutation operator used in this algorithm is designed to mutate the platforms in the product family in order to

increase the searching quality of the GA code for exploring various levels and configurations of commonality. First,

for each design variable, a random number between 0 and 1 is generated. If its value is less than the user-specifiedmutation probability, then the corresponding design variable in the product family is mutated. In mutation, a new

random number between 0 and 1 is generated. If its value is less than 0.5, then the corresponding design variable in

the product family is set as distinct in each product, and the algorithm mutates that design variable for each product

in the design variable chromosome according to the polynomial mutation operatorand modifies the commonality

chromosome accordingly. Otherwise, the design variable is made common among all products: A random number is

generated and passed to the corresponding genes in the design variable chromosome, and the commonality

chromosome is modified as well.

D. Commonality Objective FunctionIn order to have the MOGA find the optimal platform configuration, an objective function for measuring the

commonality for each family of products is added to the set of performance objective functions. Several metrics for

measuring the commonality degree in product families have been proposed reflecting various commonality benefitsbased on companys focus and standpoint. Khajivarad and Michalek42 argue that the commonality index (CI),

introduced by Martin and Ishii43, captures the cost benefits of commonality better than prior metrics used in product

family optimization, and we adopt it here as the commonality objective function. CIranges between 0 and 1 and is a

measure of unique parts; that is, a higher value indicates the whole product family was made with a fewer number of

unique parts: For a product family withp products each with n components CIcan be found as follows:

)( 11

=

pn

nuCI (1)

where u represents the total number of distinct components in the product family. By defining Ni as the number ofdistinct integers for the ith design variable in the commonality chromosome, Eq. (1) can be reformulated as follow:

)(

)(

1

1

1 1

==

pn

N

CI

n

i

i

(2)

Using the above definition, calculation of the commonality objective function uses only the commonality

chromosome while the product performance-related objectives use only the design variable chromosome; this is a

key feature that enables decomposition of the proposed GA, as discussed next.

IV. Decomposition and Parallelization of the Multi-Objective GAAforementioned modifications to the original NSGA-II code make it convenient for optimizing the product

family problem with generalized commonality; however, this algorithm is still only practical for problems with a

relatively small number of design variables and variants. The commonality generalization also increases this

complexity, making the algorithm inefficient in dealing with high-dimensional problems. To address this scalability

limitation, we propose a decomposition of the original all-in-one formulation (). The method involves allocating

the commonality chromosome to an upper-level GA and decomposing the design variable chromosome into its

product variants, where each variant is allocated to one of the lower-level sub-GAs. Commonality constraints are

imposed from the upper-level GA to lower-level ones. Using this formulation, the dimensionality of each lower-

level GA remains constant as more variants are added to the product family. The core advantage of this

decomposition is that selection of individuals in the population for producing offspring is made with respect to the


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fitness of a subsetof the full product family, rather than the entire product family chromosome. This property can

improve performance dramatically because, for example, a product family with some high performing variants need

not have a high fitness over the entire product family in order to pass on information from its high performing

variants to the next generation. In the all-in-one GA, selection of product families from the population is made

with respect to the fitness value of those families; in contrast, the decomposed GA involves 1) selection of sub-

chromosomes based on their sub-fitness values for producing offspring and 2) coordination of the sub-GAs after

each generation to select the subset of product families from the joint parent-offspring population that will advance

to the next generation. Some information on the performance of the overall family is also included in the fitness

calculation of each sub-GA in order to avoid divergence in offspring generation, but because each sub-GA does not

select on the basis of the fitness of the entire product family, but rather on the basis of a sub-fitness value, each sub-

GA can carry over features of high-performing subsets of the full product family chromosome to the next

generation. Moreover, due to the parallel nature of this decomposed method, each sub-GA can be executed on a

separate processor using the MPI (Message Passing Interface) library for sending and receiving data among nodes

after each generation. Since communication cost is negligible compared to computational cost, and performance is

improved due to decomposition, we obtain a high speedup in computational time through parallel processing.

Figure 3. Decomposed Multi-Objective GA Model for Product Family Design

The general structure of the proposed model is shown in Figure 3Error! Reference source not found.. This

figure shows the case of three products; however, the approach can easily be generalized to any number of products.

The steps of the algorithm proceed as follows:

Step 1. Initial populations are created in the upper-level and lower-level GAs independently. According to the

commonality chromosomes, the upper-level GA sends the required data to each lower-level GA, which


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modifies its design variable chromosomes to maintain consistency with the corresponding commonality

chromosomes.

Step 2. The commonality metric, Eq. (2), and individual variant performance objectives are calculated in the upper-

level and lower-level GAs, respectively. The upper-level GA sends the commonality metric to all lower-

level GAs, which are included in the fitness function of each in addition to the product performance

objectives. Each lower-level GA also returns performance deviations to the upper-level GA, and these are

summed across variants to form the overall performance objective functions. The passing of these sub-

fitness values serves to augment fitness values in the individual GA with additional information about the

performance of the corresponding family to improve selection for producing offspring.

Step 3. Using the selection, crossover and mutation operators, offspring populations are generated in all GAs. The

fitness function used for selection in each lower-level GA involves two objectives: the performance of the

corresponding variant and the commonality metric for the variants family. The fitness function used for

selection in the upper-level GA involves the fitness of the entire family (i.e.: the commonality metric and

the sum of performance deviations for all variants). Crossover and mutation operators at the product level

are the same as the sequential version except that the tasks are divided among different processors. For

example, the upper-level GA applies the two-dimensional binary crossover operator to the commonality

chromosome while lower-level GAs use simulated binary crossover restricted by commonality constraints

received from the upper-level. In case of mutation, the upper-level GA decides which design variablesshould me mutated, i.e., which variables should become common or distinct among the products, and

passes this data to the lower-level GAs so that they can mutate the individuals accordingly. The upper-level

GA sends the modification data to all lower-level GAs to make all populations consistent, and each lower-

level GA passes back the fitness value for its performance objective.

Step 4. The upper-level GA combines the parent and offspring population and applies non-dominated sorting to

select the best half as the new generation that will define the new population in all lower-level GAs as well.

Step 5. If the generation number is equal to the maximum generation number, the algorithm is terminated;

otherwise, the process is repeated from Step 2.

V. General Aviation Aircraft ExampleTo demonstrate the proposed approach, we design a family of three general aviation aircraft (GAA), an example

that was first introduced by Simpson, et al.44

and later used by Simpson and DSouza7

in their aforementioned GAimplementations. In this example, a family of three aircraft accommodating 2, 4, and 6 people is optimized. For the

purpose of this example, a GAA is defined as a fixed wing, single-engine, single pilot propeller-driven aircraft for 2,

4 and 6 passengers that can cruise at 150 to 300 knots and has a range of 800 to 1000 miles. The design challenge is

to determine the best values of the key variables for the fuselage, wing, and engine to satisfy a variety of

performance and economic requirements. The performance parameters for an aircraft with a particular set of input

variables are obtained from the General Aviation Synthesis Program (GASP) output. GASP, developed by NASA in

1978, is a synthesis and analysis program that facilitates parametric studies of small aircraft 43. It is specially suited

to analyze and study the performance characteristics of small fixed-wing aircraft having a single piston engine, fixed

pitch propeller, twin turboprop/turbofan powered business or transport type aircraft. GASP uses an input file

configured by the user and creates an output file listing various parameters for a specific aircraft produced through

computations and interactions between the sub-modules.

A. GAA Problem FormulationTable 1 summarizes the design variables and their respective bounds used in this study. Performance and

economic targets and constraints for each aircraft are listed in Tables 2 and 3, respectively. Denoting each product

target by Tj and the corresponding GASP response values by Rj and assuming the performance of all products are

equally important for us, the goal is to minimize the sum of deviations of responses from target values for the family

of products. Hence, for the non-decomposed GA, we can formulate the optimization problem with two objective

functions as shown in Eq. (3).


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Table 1. Design Variables and Bounds for GAA Problem

Variable Description Lower Bound Upper BoundWGS: Wing Loading, (lb/ft2) 20.0 25.0EMCRU: Design Cruise Speed, (mach) 0.20 0.45WS: Seat Width, (in) 15.0 20.0WAS : Aisle Width, (in) 17.0 20.0AR : Aspect Ratio 7.0 10.0SAH : Horizontal Tail Location on Vertical Tail 0 1TCR : Wing Root Thickness to Chord Ratio 0.10 0.20TCT : Wing Tip Thickness to Chord Ratio 0.10 0.20TCHT : Horizontal Tail Root Thickness to Chord Ratio 0.08 0.15ELODT : Length to Diameter Ratio of Tail Cone of Fuselage 3 4DPROP : Propeller Diameter, (ft) 5 7AF : Activity Factor per Blade 80.0 100.0

variantsofnumber:

ProducttheofResponse

ProducttheofTarget

7):exapmleGAA(inTargetsofNumber

3):ExampleGAA(inproductsofNumber

12):ExampleGAA(iniablesdesign varofNumber

1

1(

1

111

thth

thth

1

2

1 1

1

i

ij

ij

T

n

i

i

p

i

n

j

ijij

T

N

ijR

ijT

n

p

n

pn

N

f

RTnp

fT

:

:

:

:

:

)(

)

=

=

=

= =

(3)

Table 2. Targets for the GAA Problem

Targets 2-Seat 4-Seat 6-Seat

1. Aircraft Fuel Weight (lbs) 450.0 400.0 350.02. Aircraft Empty Weight (lbs) 1900.0 1950.0 2000.03. Direct Operating Cost ($/hr) 60.0 60.0 60.04. Purchase Price ($ in 1970) 41,000 42,000 43,0005. Lift to Drag Ratio 17.0 17.0 17.06. Cruise Speed (kts) 200.0 200.0 200.07. Range (nm) 2500.0 2500.0 2500.0

Table 3. Constraints for the GAA Problem

Constraints 2-Seat 4-Seat 6-Seat

1. Maximum Take-off Noise (db) 75.0 75.0 75.02. Maximum Direct Operating Cost ($/hr) 80.0 80.0 80.03. Maximum Ride Roughness Coefficient 2.0 2.0 2.0

4. Maximum Aircraft Empty Weight (lbs) 2200.0 2200.0 2200.05. Maximum Aircraft Fuel Weight (lbs) 450.0 475.0 500.06. Minimum Flight Range (nm) 2000.0 2000.0 2000.0

Constraints are handled using an exterior penalty method, which in case of multi-objective optimization should

be added to all of the objective functions (in our case f1 and f2). Therefore, the new objective functions are

formulated as follows:


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>>=


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4a. Non-Decomposed GA 4b. Parallel GA

Figure 4. Comparison of Resulting Pareto Fronts for GAA Problem

Design variables and deviation values for each point located in the Pareto optimal front (see Figure 4b) are listed

in Table 4. The concept of generalized commonality can be seen from this table, i.e., all points (except the solution

with 100% commonality) in the Pareto optimal front share variable values with a subset of the product family. The

restricted approach of allowing only all-common or all-distinct is not capable of finding these solutions. This shows

the effectiveness of the generalized two-dimensional commonality chromosome.

VI. Closing Remarks and Future WorkIn this paper, we proposed a single-stage approach for solving the joint product family optimization problem

using a novel decomposed MOGA formulation with a generalized commonality chromosome. The augmented

chromosome representation introduced by Simpson and Dsouza7 was generalized to address component sharing

among various subsets of products. Next, in order to improve the scalability of the proposed approach, the originalall-in-one MOGA was decomposed into a two-level optimization problem in which the upper-level GA finds the

optimal platform configuration while each lower-level GA optimizes one individual product in the family. A family

of three general aviation aircraft was optimized using both the all-in-one (non-decomposed) and decomposed

formulations for an equal number of generations. The Pareto optimal front found by the parallel algorithm is both

well-converged and well-distributed along the entire commonality region, and it dominates all solutions of the all-in-

one algorithm. Moreover, existence of component-sharing among various subsets of individual products in the

optimal families reveals the importance of generalizing the commonality metric. Future work entails applying the

approach to a larger product family to further test scalability and measuring the speedup due to both decomposition

and parallel processing. Moreover, a more systematic way for generating the generalized commonality chromosome

that ensures a proper distribution for the commonality probability density function should be investigated. This

feature becomes critical in finding a well-distributed Pareto front when a family with a large number of products is

considered.


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Table4. Design Variables and Deviation Values for Pareto Frontier in Figure 5b

Variables

Point 1

2-seat 4-seat 6-seat

Point 2


Point 3


0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24

9.69 9.69 9.69 9.21 9.21 9.21 9.77 9.77 9.77

5.20 5.20 5.20 5.08 5.08 5.08 5.20 5.20 5.20

24.40 24.40 24.40 24.61 24.61 24.61 24.28 24.28 24.28

95.49 95.49 95.49 91.36 91.36 91.36 85.78 85.78 85.7817.60 17.60 17.60 15.45 17.22 17.22 17.45 18.64 18.64

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

17.10 17.10 17.10 18.39 18.39 18.39 19.51 19.51 19.51

0.14 0.14 0.14 0.10 0.10 0.10 0.14 0.14 0.14

0.12 0.12 0.12 0.18 0.18 0.18 0.12 0.12 0.12

0.08 0.08 0.08 0.08 0.08 0.08 0.15 0.08 0.08

WGSEMCRUWSWAS

ARSAHTCRTCTTCHTELODTDPROPAF 3.99 3.99 3.99 3.88 3.88 3.88 4.00 4.00 4.00

Deviation(%) 2.0594 1.5953 3.3556 1.6340 1.1539 2.9594 1.0380 0.9957 2.8142

Commonality 100% 95.8% 91.7%

Variables

Point 4


Point 5


Point 6


0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24

9.74 9.74 9.74 9.77 9.77 9.77 9.45 9.45 9.98

5.19 5.19 5.19 5.20 5.20 5.20 5.13 5.13 5.35

24.99 24.99 24.99 24.29 24.29 24.29 24.61 24.61 24.61

95.91 95.91 95.91 85.66 85.66 85.66 91.78 91.78 91.7816.35 17.66 19.61 17.45 18.68 18.68 15.74 17.19 19.02

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

17.77 17.77 17.77 19.56 19.56 19.56 17.81 17.81 17.81

0.11 0.11 0.11 0.14 0.14 0.10 0.10 0.10 0.10

0.18 0.18 0.18 0.12 0.12 0.18 0.18 0.18 0.18

0.15 0.08 0.08 0.15 0.08 0.08 0.15 0.08 0.08

WGSEMCRUWSWAS

ARSAHTCRTCTTCHTELODTDPROPAF 3.95 3.95 3.95 4.00 4.00 4.00 4.00 4.00 4.00

Deviation(%) 1.0658 0.8636 2.0841 1.0230 1.0170 1.4700 1.0390 0.8880 1.6040

Commonality 87.5% 83.3% 79.2%

VariablesPoint 7

2-seat 4-seat 6-seatPoint 8



0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24

9.75 9.75 10.00 9.08 9.54 10.00 9.07 9.47 10.00

5.19 5.19 5.40 5.08 5.08 5.49 5.06 5.19 5.40

24.98 24.98 24.55 24.94 24.55 24.94 24.95 24.57 24.82

95.88 95.88 98.05 86.19 86.19 86.19 89.03 89.03 89.03

16.34 17.62 19.31 15.26 17.16 19.07 15.26 17.15 19.02

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

18.53 17.50 18.53 17.30 18.5019.0019.5

618.50 18.50 18.50

0.11 0.11 0.11 0.10 0.10 0.10 0.10 0.10 0.10

0.18 0.18 0.18 0.18 0.18 0.20 0.18 0.18 0.20

0.15 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08

WGSEMCRUWS

WASARSAHTCRTCTTCHTELODTDPROPAF

3.94 3.94 3.94 3.96 3.96 3.96 3.86 3.86 4.00

Deviation(%) 1.0761 0.8266 1.1504 0.9630 0.8210 1.0560 0.9314 0.8537 1.0150

Commonality 66.7% 62.5% 58.3%

VariablesPoint 10




0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.25

9.07 9.54 10.00 9.81 9.82 10.00 9.08 9.79 9.68

5.06 5.10 5.34 5.17 5.31 5.31 5.11 5.35 5.4724.93 24.56 24.58 24.34 24.26 24.55 24.94 24.26 24.23

87.03 87.03 87.03 87.65 82.60 90.00 87.00 83.03 80.13

15.26 17.15 18.95 17.31 18.68 18.92 15.43 18.67 19.07

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

18.56 19.06 19.48 17.92 18.20 17.92 18.08 19.00 18.15

0.10 0.10 0.10 0.14 0.14 0.10 0.10 0.14 0.10

0.18 0.18 0.19 0.12 0.12 0.19 0.18 0.12 0.18

0.08 0.08 0.08 0.15 0.08 0.08 0.09 0.08 0.08

WGSEMCRU

WSWASARSAHTCRTCTTCHTELODTDPROPAF 3.99 3.99 3.99 3.98 3.98 3.96 3.91 3.98 3.97

Deviation(%) 0.9601 0.7774 0.9955 0.9210 0.8406 0.9625 0.8450 0.8040 0.9900

Commonality 54.2% 37.5% 25.0%


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Acknowledgments

This work is supported in part by the Pennsylvania Infrastructure Technology Alliance, a partnership of

Carnegie Mellon, Lehigh University, and the Commonwealth of Pennsylvania's Department of Community and

Economic Development (DCED). Dr. Simpson also acknowledges support from the National Science Foundation

under CAREER Award No. DMI-0133923. Any opinions, findings, and conclusions or recommendations presented

in this paper are those of the authors and do not necessarily reflect the views of the sponsors.

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A Decomposed Genetic Algorithm for Solving the Joint Product Family Optimization Problem

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